Let \(d\)  be the degree of \(Z\)  over \(\mathcal {A}\) . The isomorphism \([d]^* Z_{k(S)} \cong ( [d]^*Z ) \times _S k(S)\)  and the assumption that \(Z_{k(S)} \rightarrow \mathcal {A}_{k(S)}\)  is a (PB)-cover show that the generic fibre of \([d]^*Z \rightarrow \mathcal {A}\)  is geometrically irreducible. Since \(Z\)  is normal and \(S\)  is of characteristic zero, the scheme \(Z_{k(S)}\)  is normal. By spreading out (see [1]} and [2]}), there is a dense open subscheme \(U\)  of \(S\)  such that, for all \(u \in U\) , the scheme \([d]^*Z_u\)  is normal and geometrically irreducible over \(k(u)\) . In particular, by Lemma REF , for every \(u \in U\) , the cover \(Z_u \rightarrow \mathcal {A}_u\)  of degree \(d\)  is a (PB)-cover, as required.
